基于可分凸优化的图像分解和波前重建的模型与算法研究

Sparse optimization has recently received wide attention from various areas such as imaging sciences, biomedicine, astronomy and space sciences. Among important topics of sparse optimization are image decomposition and wavefront reconstruction which play significant roles in many specific fields such as pattern recognition, materials analysis and space explorations. Typically, the existing models and algorithms for handling image decomposition and wavefront reconstruction have some limitations. For instance, the models for image decomposition are inapplicable to decompose a target image with the hybrid corruptions, e.g., convolution, missing pixel values, low-resolution and additive noise, whilst the models for wavefront reconstruction are incapable of providing solution with high accuracy. Moreover, the computational efforts for solving those models are numerically intensive because of the nonsmooth objective functions and ill-posed linear operators. Researchers in these fields are devoted to some challenging problems such as generalizing the applicable range of images for these two tasks, developing more efficient numerical solvers, and improving the accuracy of solutions. In this project, we aim at extending our previous research results on image decomposition and wavefront reconstruction in the following aspects: (1) devising image decomposition models for separating simultaneously a target image with hybrid corruptions; (2) devising wavefront reconstruction models for reconstructing high-resolution wavefront gradient by a sequence of low-resolution frames; and (3) developing efficient and robust numerical algorithms for separable convex optimization which favor parallel computing platform, and applying them to solve the proposed models on parallel computer system.

近几年来，作为最优化领域的研究热点，稀疏优化被广泛应用于图像科学、生物医学和空间科学等领域。其中，在模式识别、材料分析以及空间探测中有着重要实际意义的图像分解和波前重建问题是稀疏优化的重要研究内容。目前，图像分解和波前重建问题的数学模型所能处理的图像类型有限，数值求解过程的计算量偏大，解的精度较低。基于我们的前期工作，本项目旨在进一步拓展图像分解和波前重建问题的可分凸优化模型，建立：（1）能处理缺损图像（同时具有模糊、信息缺失、低分辨率和噪声等）的图像分解模型；（2）能获取高精度波前梯度、位相和点扩散函数的波前重建模型；（3）设计问题驱动的处理可分凸优化的并行算法，并用其处理（1）-（2）中的数学模型，最终实现图像分解和波前重建问题在并行机群上的快速、鲁棒求解。

本项目以设计求解可分凸优化问题的算法为基础，改进了图像分解问题和波前重建问题的数学模型，利用一阶可分凸优化算法快速求解。本项目的主要研究成果为：(1)算法方面。设计了三个问题驱动的求解可分凸优化的一阶方法，分别是序列迭代格式的拉格朗日乘子法、适用于并行机群的交替方向法、带临近点项的交替极小化法；(2)图像分解模型的改进。建立了基于广义全变差和矩阵低秩的图像分解模型，通过利用模型本身的可分性，实现了破损图像的质地分离。而且把图像分解思想应用到纺织业中的布料瑕疵检测问题、生物信息学中的扫描图像的结构噪声去除问题和细胞团的向量场的估计问题等；(3)波前重建模型的改进。设计了基于相位和紧框架的波前重建模型，从而保证解的唯一性质，重建出的波前有更高的精读。